What is the curved version of the Tits fibration for $G_2$?

Question

Let $\require{AMScd}$ \begin{CD} G_2/(P_1\cap P_2) @= G_2/(P_1\cap P_2)=:\mathbb{I}\\ @V \lambda V V @VV \pi V\\ \mathbb{Q}_5:=G_2/P_1 && G_2/P_2=:\mathbb{N}_5 \end{CD} be the Tits fibration of the exceptional Lie group $G_2$ (see Landsberg and Manivel, § 4.1, for the general definition, and Bryant, pag. 12, for the particular example of $G_2$).

I recall that $\mathbb{I}$ is a $\mathbb{P}^1$-bundle over the 5-dimensional contact manifold $\mathbb{N}_5$, such that, for any $p\in \mathbb{N}_5$, the fibre $\pi^{-1}(p)$ is a twisted cubic in $\mathbb{P}\mathcal{C}_p\cong\mathbb{P}^3$ - the projectivised contact space at $p$. In particular, we can think of $\mathbb{I}$ as a sub-bundle of $\mathbb{P}\mathcal{C}$, which in turn is a sub-bundle of $\mathbb{P}T\mathbb{N}_5$.

Let now $\gamma\subset\mathbb{N}_5$ be a curve, and denote by $T\gamma\subset T\mathbb{N}_5|_\gamma$ its (rank-one) tangent bundle. Since each fibre of $T\gamma$ is a tangent line to $\mathbb{N}_5$, it makes sense to regard $T\gamma$ as a curve in $\mathbb{P}T\mathbb{N}_5$.

Then one can prove that $$\mathbb{Q}_5=\{\gamma\mid T\gamma\subset\mathbb{I} \}\, .\quad\quad (^*)$$

Now $\mathbb{N}_5$ is just the homogeneous model of Cartan geometries of type $(G_2,P_2)$, and I'm wodering to what extent the Tits fibration can be recast in a non-flat setting.

More pecisely, let $M$ be a Cartan geometry of type $(G_2,P_2)$. In particular, there exists a $\mathbb{P}^1$-bundle $\mathbb{I}\to M$, such that $\mathbb{I}\subset\mathbb{P}TM$. This means that definition $(^*)$ can be repeated verbatim, viz. $$ Q_M:=\{\gamma\textrm{ curve in }M\mid T\gamma\subset\mathbb{I}\}\, . $$

QUESTION. It is obvious that $Q_M$ is a Cartan geometry of type $(G_2,P_1)$? If yes, what is a good reference for that?

This is related to a previous question, which led to interesting discussions, but it is still unanswered (see also this one).

Answer 1

This is an instance of the general construction of correspondence spaces and twistor spaces as described in my article MR2139714 and in Chapter 4 of the book of Jan Slovak and myself on parabolic geometries. A Cartan geometry of of type $(G_2,P_2)$ can be described as a 5-dimensional contact manifold $(M,H)$ together with an auxilliary rank-two bundle $E\to M$ such that $H\cong S^3E$. (This is an equivalent way to phrase the twisted cubic you mention.) Now the correspondence space $N:=\mathcal CM$ is the total space of the projectivization of $E$ and thus a circle bundle over $M$. Denoting by $\mathcal G\to M$ the Cartan bundle, it is easy to see that $N\cong \mathcal G\times_{P_2}(P_1\cap P_2)\cong\mathcal G/(P_1\cap P_2)$. This shows that there is a canonical projection $\pi:\mathcal G\to N$, which is a principal fiber bundle with structure group $P_1\cap P_2$. The canonical Cartan connection $\omega$ associated to the original geometry by definition also is a Cartan connection on this bundle, thus defining a parabolic geometry of type $(G_2,P_1\cap P_2)$ on $N$.

This shows that $TN\cong\mathcal G\times_{P_1\cap P_2}(\mathfrak g/(\mathfrak p_1\cap\mathfrak p_2))$ and hence $\mathfrak p_1/(\mathfrak p_1\cap\mathfrak p_2)$ defines a line subbundle in $TN$. For this you can form a local leaf space $Z$, which is called a twistor space, so locally (on $N$) you have a double fibration picture. But you won't get a Cartan geometry on $Z$ unless the initial geometry is locally flat. I am not entirely sure how things look like in the example you are considering but I think that you can identify the pre-image of $Z$ in $\mathcal G$ with an open subset in a principal $P_1$-bundle over $Z$. But carrying $\omega$ over to this bundle, the result will be $P_1$-equivariant (thus defining a Cartan connection) if and only if the initial geometry is locally flat. So there is no way to directly get a Cartan geometry in non-flat cases.

Of course, one could try to directly construct a rank two-distribution on a local leaf space $Z$ from the given data (and it would be easy to find candidates), maybe imposing some restrictions on the curvature of the initial geometry. I have not tried that, but I wouldn't be very optimistic.